Numerical Simulations of Penetration and Overshoot in the Sun

Tamara M. Rogers

Astronomy and Astrophysics Department, University of California, Santa Cruz, CA 95064

trogers@pmc.ucsc.edu

Gary A. Glatzmaier

Earth Sciences Department, University of California, Santa Cruz, CA 95064

C.A. Jones

Department of Applied Math, University of Leeds, Leeds LS2 9JT, UK

ABSTRACT

We present numerical simulations of convective overshoot in a two-

dimensional model of the solar equatorial plane. The model equations are solved

in the anelastic approximation with enhanced thermal conductivity and viscosity

for numerical stability. The simulated domain extends from 0.001 R to 0.93 R,

spanning both convective and radiative regions. We show that convective pen-

etration leads to a slightly extended, mildly subadiabatic temperature gradient

beneath the convection zone, spanning approximately 0.05 Hp, below which there

is a rapid transition to a strongly subadiabatic region. A slightly higher tempera-

ture is maintained in the overshoot region by adiabatic heating from overshooting

plumes. This enhanced temperature may partially account for the sound speed

discrepancy between the standard solar model and helioseismology. Simulations

conducted with tracer particles suggest that a fully mixed region exists down to

at least 0.684 R.

Subject headings: convection: overshoot,mixing

1. Introduction

One of the main unsolved problems in all of stellar evolution theory is the treatment

of convective-radiative boundaries. In the Sun, understanding convective overshoot is cru-

cial in determining properties of the solar tachocline and the solar dynamo. Helioseismic

– 2 –

observations have shown that the differential rotation observed at the solar surface persists

throughout the convection zone (Thompson et al. 1996). More surprisingly, the p-mode

splittings showed that over a very thin radial extent the rotation profile changes from differ-

ential in the convection zone, to solid body in the radiative interior. The unresolved radius

over which this transition occurs has been named the tachocline (Spiegel & Zahn 1992).

Whether this region is quiescent and mostly devoid of overshooting motions or is violent and

constantly bombarded by plumes is an unanswered question, the solution to which may help

constrain theoretical models for the tachocline.

Understanding the tachocline is not only important for comprehending the internal

rotation of the Sun; it is crucial for understanding the dynamo process. In classic mean field

theory, the dynamo process is explained in two steps: poloidal field is sheared into toroidal

field by differential rotation (the Ω effect), toroidal field then buoyantly rises and, because

of Coriolis forces is twisted back into a poloidal field (the α effect). These processes were

initially postulated to occur in the convection zone. However, it became clear some time ago

(Parker 1975) that magnetic field would become buoyant and be quickly shredded within

the turbulent convection zone. During the ensuing years it was proposed that the storage

and amplification of field could occur in the stably-stratified overshoot region (Spiegel &

Weiss 1980). The tachocline, with its strong differential rotation would provide the ideal

site for storage and amplification of the field. To explain how the αΩ dynamo would work

in this scenario Parker (1993) proposed the interface dynamo, which places the Ω effect in

the stable region beneath the convection zone, while keeping the alpha effect in the bulk

of the convection zone. If the interface dynamo is to work, the overshoot region must play

two crucial roles in the dynamo cycle: (1) the stable stratification allows field, which is

transported into the region by overshooting plumes (Tobias et al. 1998), to be stored there

on the solar cycle timescale and (2) the strong shear in this region must convert poloidal field

into toroidal field. Understanding the precise nature of this overshoot region, the amplitude

of the subadiabaticity and its depth, is crucial for understanding the efficiency with which

field can be pumped into the overshoot region, the magnitude of the field capable of being

stored there and the timescale on which it could be stored.

The problem of overshoot is not confined to the Sun. Most stages of stellar evolution

are affected. The Lithium depletion in some main sequence stars may be explained by

convective overshoot. In more massive stars with convective cores, overshoot can lead to

increased central hydrogen abundance, and therefore longer main sequence lifetimes, which

in turn affects isochrone fitting and age predictions for stellar clusters. Dredge up during the

Asymptotic Giant Branch (AGB) phase can lead to the surface enrichment of material from

the core and to the presence of 13C necessary for s-process nucleosynthesis (Herwig 2000).

Convective overshoot can affect the energy production and nucleosynthesis of classical novae

– 3 –

by enriching the accreted layer with underlying white dwarf material (Woosley 1986).

Clearly, overshoot is an important physical process which must be explained if stellar

evolution is to be better understood. Many analytic and numerical theories have been pro-

posed, but no clear consensus has been reached. Early analytic models (Shaviv & Salpeter

1973, van Ballegoojen 1982, Schmitt, Rosner & Bohn 1984) predicted extended adiabatic

regions and early numerical experiments concurred (Hurlburt et al. 1994). More recent

analytic results (Zahn 1991, Rieutord & Zahn 1995, Rempel 2004) model the penetration

using plume dynamics and agree that the extent and nature of the overshoot depends crit-

ically on the filling factor of the plumes at the base of the convection zone as well as the

flux. It also appears that the inclusion of upflow-downflow interaction, and the prescription

for it, affects the penetration depth (Rieutord & Zahn 1995, Rempel 2004). Numerous nu-

merical experiments have also been conducted. Two-dimensional models (Massaguer et al.

1984, Hurlburt et al. 1986, Hurlburt et al. 1994, Rogers & Glatzmaier 2005a) have been

conducted to determine the nature of convective overshoot as well as dependencies of the

penetration depth on other physical variables such as stability of the stable region and the

input flux. Three-dimensional simulations have also been conducted (Chan & Sofia 1989,

Singh et al. 1998b, Saikia et al. 2000, Brummell et al. 2002). While the earlier three di-

mensional models recover the scaling relationship between penetration depth and flux (rms

velocity) determined analytically and in two dimensional numerical experiments, the more

recent numerical simulations find that this scaling law breaks down at higher levels of the

stability of the underlying radiative region (Brummell et al. 2002).

All analytic predictions make some crude assumptions, while all numerical simulations

employ far from realistic solar parameters. The numerical simulations presented in this work

still can not reach the level of turbulence that surely exists in the Sun. However, in this work

we have taken a few steps forward by using a realistic thermal profile, including rotation,

employing a more realistic geometry and including most of the convective and radiative

regions. In section 2 we describe our numerical model and equations; in section 3 we discuss

some of the features of basic convective penetration. In section 4 we present estimates for

the depth of the penetrative convection before our reference state model is evolved. Section

5 details the evolution of the mean thermal profile and section 6 compares these results with

previous analytic and numerical models. In section 7 we discuss mixing of species as modeled

by tracer particles.

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2. Numerical Model

2.1. Equations

The numerical technique used here is identical to that in Rogers & Glatzmaier (2005b).

We solve the Navier Stokes equations in the anelastic approximation (Gough 1969) for per-

turbations about a mean thermodynamic reference state. The equations are solved in 2D

cylindrical geometry, with the computational domain extending from 0.001 R to 0.93 R,

representing the equatorial plane of the Sun. The radially (r) dependent reference state is

taken from a standard 1D solar model (Christensen-Dalsgaard private communication). The

radiative region spans from 0.001R to 0.718R, while the convection region occupies the

region from 0.718R to 0.90R and for numerical ease an additional stable region is included

from 0.90R to 0.93R. Excluding 0.1 % of the solar radius in the core should have little

effect, as gravity waves reflect when their frequency approaches the Brunt-Vaisala frequency,

which vanishes at the core.1

For numerical simplicity we solve the curl of the momentum equation, the vorticity

equation:

∂ω

∂t+ (v · ∇)ω = (2Ω + ω)hρvr − g

Tr

∂T

∂θ− 1

ρTr

∂T

∂r

∂p

∂θ+ ν∇2ω (1)

where, ω = ∇ × v is the vorticity in the z direction and v is the velocity, comprised of

a radial component, vr, and a longitudinal component, vθ. T is the temperature, ρ is the

density, p is the pressure, and g represents gravity. Overbars denote prescribed reference

state variables taken from a 21st order polynomial fit to the 1D solar model. These values are

functions of radius and slow functions of time when the reference state is allowed to evolve

(see section 2.2). Variables without overbars are the time dependent perturbations which

are solved for relative to the reference state. Derivatives of the reference state values are

calculated from analytic derivatives of the polynomial expansions. The viscous diffusivity, ν,

is radially dependent and defined such that the dynamic viscosity, νρ is constant (initially).

The rotation rate, Ω, is set equal to the mean solar rotation rate, 2.6×10−6 rad/s.

The energy equation is solved as a temperature equation:

∂T

∂t+ (v · ∇)T = −vr(

∂T

∂r− (γ − 1)Thρ)+

1However, we do not fully understand the properties of nonlinear waves under internal reflection. It ispossible that internal reflection is physically distinct from reflection off of a hard boundary.

– 5 –

(γ − 1)Thρvr + γκ[∇2T + (hρ + hκ)∂T

∂r]+

γκ[∇2T + (hρ + hκ)∂T

∂r] +

Q

cv(2)

In Equation (2), κ is the (radially dependent) thermal diffusivity, (γ − 1) = (dlnT/dlnρ)ad,

hρ = dlnρ/dr and hκ = dlnκ/dr. These last two, hρ and hκ, represent inverse scale heights.

Q represents physics included in the 1D standard solar model, but not included here, which

maintains the initial reference state temperature gradient. In the convection zone, Q is the

divergence of the mixing length flux which, together with the second to last term in (2)

accounts for the convergence of the total reference state flux through the system. Initially,

this sum is set to zero, so that the initial reference state temperature is time independent.

In this model, similar to Rogers & Glatzmaier 2005b, we use the temperature as our

working thermodynamic variable, rather than entropy. We do this to avoid the inward heat

flux that would accompany a positive entropy gradient and large turbulent diffusivity in the

stable, radiative interior.

We calculate the pressure term in (1) by solving the longitudinal component of the

momentum equation:

1

ρr

∂p

∂θ= −∂vθ

∂t− (v · ∇v)θ + ν[(∇2v)θ − hρ

3r

∂vr

∂θ] (3)

These equations are supplemented by the continuity equation in the anelastic approximation:

∇ · ρv = 0 (4)

The convective and radiative regions are set up by taking the subadiabaticity defined

as:

∆∇T = (γ − 1)hρT − ∂T

∂r≈ ((

∂T

∂r)ad − ∂T

∂r) (5)

from the 1D model and specifying the superadiabaticity in the convection zone as the con-

stant value 10−7 K/cm. The thermal diffusivity, κ is given by the solar model, multiplied by

a constant factor, κm for numerical stability:

κ = κm16σT

3

3ρ2kcp

(6)

where σ is the Stefan-Boltzman constant, k is the opacity (taken from the 1D solar model)

and cp is the specific heat at constant pressure. The multiplying factor, κm, is 105 and

therefore, the convective heat flux is 105 larger than the solar value. This produces the

– 6 –

proper radial profile of the radiative diffusivity, albeit increased by a large factor for numerical

stability; this is a “turbulent” diffusivity.

Since both ν and κ vary with height, the relevant control parameters, such as Pr (Prandtl

number = ν/κ), Ra (Rayleigh number =g∆∇TD4/νκT , where D represents the convection

zone depth) and Ek (Ekman number =ν/(2ΩD2)) vary with height. The Pr varies from

10−2BCZ (at base of the convection zone) to 0.7TCZ (at top of convection zone) and is 10−3 near

the core. The Ra varies from ≈ 108BCZto ≈ 107

TCZ and the Ekman number (Ek=ν/2ΩD2)

varies from 10−4BCZ to 10−2

TCZ .

These equations are solved using a Fourier spectral transform in the longitudinal (θ)

direction and a finite difference scheme on a non-uniform grid in the radial (r) direction.

Time advancing is done using the explicit Adams-Bashforth method for the nonlinear terms

and an implicit Crank-Nicolson scheme for the linear terms. The domain resolution is 2048

x 1500, with 620 radial levels dedicated to the radiative region. The radial resolution in

the overshoot region is 170km. The boundary conditions on the velocity are stress-free and

impermeable, while the temperature boundary condition is specified as constant temperature

gradient at the top and constant temperature at the bottom.

2.2. Updating the Reference State

After the model had run ≈ 1 year (which requires nearly 4 million timesteps), we allow

the mean reference state to evolve in response to the convection. The procedure is as follows:

(1) Half of the mean temperature, density and pressure perturbations (T (m = 0 , r),

ρ(m = 0 , r) and p(m = 0 , r)) are added to the reference state values T , ρ, p forming the

new reference state:

T new = T old +1

2T (m = 0 , r) (7)

ρnew = ρold +1

2ρ(m = 0 , r) (8)

P new = P old +1

2P(m = 0 , r) (9)

Here, m is the longitudinal wavenumber and m=0 represents the axisymmetric (mean) per-

turbation. The remainder of the mean perturbation is maintained as a perturbation so that

the model can adjust gradually.

(2) Using the new values for density and temperature, a new opacity is calculated using

Kramers Law

k(r) = C(r)ρnew(r)T new(r)−3.5 (10)

– 7 –

where C(r) is obtained by matching the original opacity profile, taken from the 1D solar

model, to a Kramer’s Law opacity, i.e. C(r)ρold(r)Told(r)−3.5 = k(r)1Dsolarmodel.

(3) The thermal diffusivity κ is recalculated with the new opacity, temperature and

density via equation (6), the multiplying factor κm is maintained.

(4) The last two terms in (2) are updated, holding Q constant.

Initially, this procedure is done at regular intervals. After the initial relatively large

changes, this procedure is done only if the mean thermodynamic perturbations become

larger than 1% of the reference state values.

3. Convective penetration and Gravity Wave generation

Turbulent convection is dominated by plumes, which form near the top of the convection

zone and descend through the bulk of the convection zone. These plumes tilt and sway and

often merge with neighboring plumes during their descent. Within the convection zone the

kinetic energy spectra varies between m−2 and m−4 depending on the radius at which it is

measured and the Ekman number. The plumes terminate their descent upon encountering

the stiff underlying stable region (Figure 1). In this region a downwelling plume adiabatically

heats relative to the subadiabatic background (hence the white “spots” seen in Figure 1 at

the base of the convection zone) and is then rapidly decelerated by buoyancy. The depth over

which the plume is buoyantly stopped depends sensitively on the stiffness of the underlying

radiative region (Hurlburt et al. 1994, Brummell et al. 2002, Rogers & Glatzmaier 2005a).

When the plumes impinge on the underlying radiative region they generate a spectrum

of gravity waves, ranging in frequency from 1µHz to 300µHz (Rogers & Glatzmaier 2005b).

Higher frequency waves set up standing modes which may ultimately be detected by he-

lioseismology. The lower frequency waves are radiatively dissipated, thereby sharing their

angular momentum with the mean flow.

4. Convective Overshoot

There are several ways to define the depth of convective overshoot. In this section

we consider only the depth to which subadiabatic overshooting motions can extend into the

stable radiative region. In the next section we consider whether these motions can extend the

adiabatic region beyond that prescribed by the reference state model. As we are concerned

with the cessation of large amplitude motion, we consider the amplitudes of kinetic energy

– 8 –

density ρ(v2r +v2

θ), vertical kinetic energy flux vr(kinetic energy density) and convective heat

flux (ρTvrcp) as a means for determining the depth of convective overshoot. The time and

horizontally averaged kinetic energy density as a function of radius is shown in Figure 2a.

The kinetic energy density drops rapidly moving into the radiative region because of the stiff

underlying stable region. The peak energy density is ≈ 5 × 108ergs/cm3, but drops to 104

ergs/cm3 at just one Hp (pressure scale height) below the convection zone 2. If we define

the depth of overshoot to be the distance between the base of the convection zone and the

height at which the kinetic energy density reaches 1% of its peak value, then that depth is

5.3×108cm, or 0.09 Hp (Hp is the pressure scale height at the base of the convection zone,

5.8 × 109cm). If instead we use 5% of the peak, the depth becomes 0.06 Hp.

An alternative, more physical way of defining the depth of convective overshoot is based

on where the kinetic energy flux changes sign. The average vertical kinetic energy flux3

(Figure 2b) is negative (downward) in the bulk of the convection zone due to descending

plumes. In this model, because of the stiffness of the underlying radiative region, the flux

changes sign well within the convective region and is positive at the base of the convection

zone. In this region descending plumes rebound, causing a net upward flux. The extent of

the region with positive vertical kinetic energy flux could be considered an overshoot depth.

This yields a smaller depth of 0.02 Hp.

Another physical measure of overshoot can be found if one considers the correlation

between temperature perturbation and vertical velocity i.e., the convective heat flux. Typ-

ically, a positive temperature perturbation (hot fluid) would give rise to a positive vertical

velocity. However, in the overshoot region, positive temperature perturbations are associ-

ated with negative vertical velocities. Therefore, in the overshoot region the quantity Tvr is

negative, as seen in Figure 2c, leading to a similar measure of the overshoot depth, 0.03Hp.

The reduction in kinetic energy density from peak is 20% for 0.03Hp penetration depth and

28% for 0.02Hp.

5. Evolving the Reference State

In order to study how the overshooting motions discussed above affect the mean ther-

mal state of the system, we allow the model to evolve in response to those motions. We can

2Note that the kinetic energy density presented here is larger than one might expect in the Sun. This isdue to increased velocities driven by a larger than solar thermal diffusivity.

3Plotted in Figure 2b is the kinetic energy flux divided by the reference state diffusive heat flux,κρcpdT/dr.

– 9 –

then determine if we find an extended adiabatic region or just subadiabatic overshoot. An

extended adiabatic region is generally referred to as “penetration depth” and distinguished

physically and colloquially from overshoot, or just subadiabatic overshoot. After the model

has run nearly one year the reference state thermodynamic variables are updated. The

horizontal average of the temperature perturbation (i.e. the m=0 mode of the Fourier ex-

pansion) is added to the reference state temperature, as described in section 2.2. Similarly,

the horizontally averaged density and pressure perturbations are added to their reference

state values. The newly formed density and temperature are used to define a new opacity

using Kramers Law (10). Using the new opacity and thermodynamic variables a new thermal

diffusivity is obtained using (6). A new reference state is thus formed. Initially, this proce-

dure is done periodically. Later, the reference state is changed only when the temperature

perturbation reaches 1% of the mean temperature at any location.

At any given instant the temperature perturbations are a small fraction of the mean tem-

perature; however, the continual evolution of the temperature, density and opacity produce

large changes in the mean thermal profile (Figure 3). In Figure 3a we show the super-

/subadiabaticity, as defined in (5), for the initial model (solid line) and our evolved model

(dotted line); in Figure 3b we show the temperature. We show only a small area around the

overshoot region so as to highlight the regions where the most change has occurred4. The

bulk of the convection zone remains superadiabatic, however, near the base of the convec-

tion zone, the region becomes slightly subadiabatic. In addition, just below the convection

zone there is a slightly extended mildly subadiabatic region, where the initial steep gradient

between super- and subadiabatic temperature gradients has evolved into a smoother tran-

sition. In this region the temperature gradient (see Figure 3b, region labeled 1) becomes

slightly steeper so that the upward diffusive heat flux can increase to compensate for the

negative convective heat flux in that region (Figure 2b). The depth of the extended mildly

subadiabatic region is roughly 0.05 Hp, if that depth is measured at the point where the

subadiabaticity reaches 1 × 10−5 (which is marked by the arrow in figure 3a). However,

the choice of this depth is arbitrary as it is unclear where the model changes from “mildly

subadiabatic” to “strongly subadiabatic”.

Convective motions are continually pumping heat into the region between labels 1 and

2 in Figure 3b. The timescale for this transfer is much shorter than the diffusive timescale

and therefore, heats up relative to the standard solar model. This mild heating causes a

flatter temperature profile at the region labeled 2 and a steeper profile at region 1. Hence,

region 1 becomes less subadiabatic and region 2 becomes more subadiabatic, which makes

4There is very little change in the deep radiative interior, nor in the bulk of the convection zone.

– 10 –

the change from nearly adiabatic to strongly subadiabatic occur over a shallower depth than

in the standard solar model.

This small difference in temperature between the standard solar model and our evolved,

hydrodynamic model is in the same sense, and at the same radius, as the standard solar

model-helioseismology sound speed discrepancy (Christensen-Dalsgaard 2002). In region 2

the maximum difference in temperature as a fraction of the original temperature (δT/T ) is

∼0.019. This difference has approximately the same amplitude and is in the same direction as

the previous standard solar model-helioseismology discrepancy, before gravitational settling

was included, but is larger than that discrepancy when gravitational settling is included. It is

possible this effect provides an additional explanation for the sound speed anomaly between

the standard solar model and helioseismology.

Because of the extended, mildly subadiabatic region seen in Figure 3, convective motions

can penetrate further into the stable region (Figure 4). In Figure 4 we see that substantial

kinetic energy density, kinetic energy flux and convective heat flux stretch further into the

stable region compared with Figure 2 (before updating the background state). Within the

convection zone, the now non-constant superadiabaticity yields kinetic energy flux and con-

vective heat flux which are not as smooth as they were previously. Key physical features, such

as a positive kinetic energy flux and negative convective heat flux, just below the convection

zone, are still apparent. Subadiabatic overshooting motions in an evolved background state

can extend down to 0.38 Hp (0.687 R)5, still significantly above the base of the tachocline

at ∼ 0.65 R. In summary, we find an extended mildly subadiabatic region down to a depth

of roughly 0.05Hp6, with subadiabatic overshoot extending further to 0.38Hp.

6. Comparison with Previous Results

A review of literature on the subject of penetrative convection yields various predictions

and no consensus. Both numerical and analytic models have been used.Early numerical

simulations in two dimensions (Hurlburt et al. 1994) found that the depth of penetration

depended primarily on the ratio of the sub- to superadiabaticity, S. These simulations found

that for low values of this ratio (1-4), the penetration depth scaled as S−1 indicating an

extended adiabatic region in agreement with analytic results by Zahn (1991) and numerical

5Note that this is as measured by the change in sign of the convective flux and is very similar to thatmeasured by tracer particles (see Section 7)

6Measured at the arrow in figure 3.6a with the stipulation that the definition of mildly subadiabatic issubjective.

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results by Singh et al. (1998) and Saikia et al. (2000). For larger values of S (between 5 and

20), the penetration scaled as dpen ∝ S−1/4. Early simulations in three dimensions (Singh,

Roxburgh & Chan 1995) also find evidence for two different scaling laws at low S (1-3) and

moderate S (5-7). However, more recent simulations of turbulent convection in 3D (Brummell

et al. 2002) found only strongly subadiabatic overshoot and no extended adiabatic region,

hence retrieving only the dpen ∝ S−1/4 scaling. The authors argue this discrepancy could be

due to the much smaller filling factor that arises in the 3D simulations or the large velocities

attained in 2D due to flywheel type motion. Our more recent simulations in 2D, using stiffer

values for the stability (closer to solar values) of the underlying radiative regions find no

extended adiabatic region (Rogers & Glatzmaier 2005a). The discrepancies between the

different numerical simulations can be traced to the reference state employed (in particular,

the value of subadiabaticity), the degree of turbulence and the dimensionality of the model.

Analytic models have been in somewhat better agreement. Early models by Shaviv

& Salpeter (1973), van Ballegoojen (1982) and Schmitt, Rosner & Bohn (1984) all found

penetration depths ranging from 20% to 40% Hp7. This appeared to be a robust solution as

they all arrived at these values using vastly different models. The work by Schmitt, Rosner

& Bohn (1984) was the first of the modern analytic solutions which model penetration using

plumes (Zahn 1991, Rieutord & Zahn 1995, Rempel 2004). All of these plume models find

that the penetration depth depends critically on the filling factor, f , that is, the fractional

area occupied by the plumes at the base of the convection zone. Schmitt, Rosner & Bohn

(1984) found that the penetration depth scaled as v 3/2 f 1/2 ; later, Zahn (1991) provided a

derivation of this scaling. He showed that:

d2pen =

3

5HpHχf

ρV 3

Ftotal(11)

demonstrating how the penetration depth depends not only on the filling factor (f ), but also

on the plume exit velocity (V), total heat flux (Ftotal) and thermal diffusivity (through Hχ).

If we compare our model directly to the expression derived by Zahn (1991) for the pen-

etration depth (11), we can isolate any fundamental differences. For instance, both Hp and

ρ are taken from the solar model in our simulations and therefore, can not be criticized as

unrealistic. While our flux is increased by 105 (κm), our velocities are also larger than one

would expect in the solar convection zone, and therefore, the ratio V 3/Ftotal is likely not

unrealistic. Finally, the fundamental complaint generally levied against numerical simula-

tions is their enhanced thermal diffusivity required for numerical stability. At issue is that

7Note that this is the extent of the extended mildly subadiabatic region, not the extent which plumesovershoot (subadiabatic overshoot).

– 12 –

this enhanced diffusivity allows the overshooting plumes to thermalize with the background

thermodynamic state more quickly than they would if the solar thermal diffusivity were

used, hence leading to lower overshoot depths. However, as is seen in (11) the quantity upon

which the penetration depth depends is Hχ, the radiative conductivity scale height. While

our simulation uses an enhanced thermal diffusivity, the radiative conductivity scale height ,

represented by Hχ is taken directly from the solar model and therefore, can not be criticized

as unrealistic. The only factor which may be criticized is the filling factor, which we admit,

may be enhanced because we are only modeling two dimensions.

Using a more sophisticated plume model Rieutord & Zahn (1995) found that when

the effect of the surrounding upflow is included in their plume model the penetration depth

decreases almost linearly with the number of plumes. When the interaction between up-

flows and downflows is neglected, the penetration depth is found to be between 0.2 Hp and

0.4 Hp, as found in previous analytic solutions, but which was inconsistent with helioseis-

mic observations (Christensen-Dalsgaard et al 1995). However, when that interaction is

included the penetration depth depends sensitively on the number of plumes. More plumes

results in more upward momentum loading by upwelling fluid. More recently Rempel (2004)

considered a semi-analytic overshoot model based on plumes. His model also includes the

interactions of plumes with upflows and finds the same behavior of decreasing penetration

depth with increased upflow-downflow interaction (see his figure 7). These results illustrate

that the penetration depth depends, rather strongly, on the upflow-downflow interaction,

which is parametrized in the best analytic models, but self-consistently calculated in numer-

ical simulations. In light of this it seems probable that at least part of the inconsistency

between numerical and analytic models lies in the simplified treatment of upflow-downflow

interaction in analytic models.

In addition to the dependence on mixing between upflows and downflows (parametrized

as α) Rempel finds that the depth of overshoot and the nature of the transition between

convective and radiative zones depends on the ratio of the total flux to filling factor:

φ = Ftotal/(fpbc(pbc/ρbc)0.5) (12)

where f is the filling factor, pbc and ρbc are the pressure and density at the base of the

convection zone, respectively. He finds that the main difference between numerical and

analytic results lies in the values of this ratio; with numerical simulations using values of

φ ∼ 10−2 (because of their increased fluxes) and analytic models employing values around

10−9. The model we present here has φ ∼ 10−3 − 10−4 depending on the filling factor of

the model. Our evolved reference state superadiabaticity (Figure 3a) resembles their model

for φ ∼ 10−4. However, it is difficult to make a direct comparison, given the dependence

– 13 –

of their model on α, although there does appear to be some degree of agreement (slightly

subadiabatic base of the convection zone, slightly extended mildly subadiabatic region).8

To recap, combined numerical and analytic models have found that the penetration

depth depends mainly on: (1) the subadiabaticity of the underlying stable region, (2) the

filling factor at the base of the convection zone and (3) the interaction of upflows and down-

flows. Low subadiabaticity leads to larger penetration depths, while high subadiabaticity

yields small penetration depths. Large filling factors lead to large penetration depths and

vice versa, depending on the number of plumes which occupy that filling factor. For a large

number of plumes the penetration depth decreases, because of enhanced upflow-downflow in-

teraction, while for a low number of plumes the penetration depth increases. Our numerical

simulations accurately represent the subadiabaticity of the radiation zone and the interaction

of upflows and downflows. However, the filling factor depends sensitively on the properties of

the convection (Ra, Re, Pr) and on the geometry and dimensionality of the flow. Resolving

the issue of filling factor remains an important issue, which should be addressed both by

analytic and numerical studies.

7. Tracer Particles

To study the effect of overshooting motions on the mixing of species, we introduce tracer

particles, which are advected by the flow. These particles are introduced after the reference

state has been evolved. We are particularly interested in estimating how deep below the

convection zone the fluid is efficiently mixed.

Five sets of 500 tracer particles were initiated at five different radii, equally distributed

in longitude (Figure 5a). Two sets were initiated in the convection zone (at 0.83 R and 0.79

R, represented in Figure 5 by red and blue points, respectively) and 3 sets were initiated in

the radiative zone (at 0.705 R, 0.690 R and 0.675 R, represented in Figure 5 by purple,

cyan and black, respectively). The yellow line represents the base of the convection zone.

Figure 5b shows the distribution of the particles after two convective turnover times. The

particles initiated within the convection zone (red and blue) are distributed throughout the

convection zone after two convective turnover times. Of the particles initiated in the stable

region many are pulled into the convection zone. The sets initiated at 0.705 R and 0.690

R both have particles within the convection zone in steady state. However, no particles

initiated at 0.675 R have made it into the convection zone on this timescale, although some

8Note that we present the super/subadiabatic temperature gradient, while they show the dlnT/dlnP, sothat the amplitudes are not similar but the profiles of super-/subadiabaticity are similar.

– 14 –

have migrated from their initial positions.

Figure 6 shows the number of particles, of the two sets initiated in the convection zone,

which are pumped into the radiation zone (any radius below 0.718 R) over two convective

turnover times. Both sets converge to having approximately 90 particles (or 18% of the

initial 500 particles) within the radiation zone at any given time. However, most of these

particles do not reach any appreciable depth below the convection zone, existing just beneath

the transition. Figure 7 shows the number of particles that started in the convection zone

and made it into the radiative region as a function of time (left) and radius (right below

the convection zone). The number of particles drops rapidly below the convection zone,

with a nearly linear drop down to 0.705 R and a slightly less rapid drop below that. The

concentrations plotted in Figure 7b are the steady state concentrations; however, the tail

of the distribution (the depth at which no particles are found) depends on the number of

particles initiated, as well as the time simulated, which is very short here.

To ascertain the effect of the number of particles as well as the time simulated (with

particles) on the minimum depth achieved by any particle we ran two additional test cases.

The initial simulation had a total of 1000 tracer particles initiated within the convection

zone, the two test simulations initiated 2000 and 4000 particles within the convection zone,

respectively (half at 0.83 R and half at 0.79 R, as in the original setup). The simulation run

with 2000 and 4000 both particles yielded a minimum depth of 0.684 R, indicating that the

initial 1000 particles were not sufficient to yield a converged result. The two test simulations

were also run longer, a total of four convective turnover times each. The minimum depth of

any particle occurred within the first convective turnover time in all cases, indicating that

the time run (in this small range) does not affect the mixing depth. However, we note that

rare, deep excursions such as those seen in three dimensional box simulations (Brummell et

al. 2002) are not seen here and therefore, a longer integration time would likely have a more

obvious effect in three dimensions.

Of the three sets of particles initiated within the radiative region, only the particles

started at 0.705 R and 0.690 R, are pulled into the convection zone. Figure 8 shows the

number of particles as a function of time, for the three sets of particles initiated within the

radiative region, which make it into the convection zone (any radius larger than 0.718 R).

As can be seen in Figure 9, nearly half of the 500 particles (230, 46%) initiated at 0.705 Rare in the convection zone in steady state. Of the particles initiated at 0.690 R, nearly 40

particles (20 % of those initiated) are within the convection zone in steady state. None of the

particles initiated at 0.675 R make it into the convection zone during these two turnover

times. The maximum radius that any particle initiated at 0.675 R reaches is 0.705 Rduring the time simulated.

– 15 –

The number of particles that are pumped down to the radius 0.690 R (1-2) is signifi-

cantly lower than the number of particles which are dredged up from this same radius (40) in

steady state (compare Figures 7 and 8). That is, there is an asymmetry between the number

of particles dredged up from a particular radius and the number of particles pumped down

to that same radius. This is because downwelling plumes in the convection zone have a small

cross sectional area and are therefore unable to entrain many particles during their descent.

However, once plumes reach the radiation zone their surface area is increased as they are

decelerated. These plumes spread transversely and are turned upward. This “scooping” mo-

tion, allows a greater number of particles to be dredged up than are pumped down. In the

Sun, where the region is completely mixed on timescales much shorter than the evolutionary

timescale, this effect is not important. However, this asymmetry could be very important in

the late stages of stellar evolution of massive stars, when the evolutionary time is closer to

the convective turnover time. This effect could also be crucial in Classical Novae or X-ray

bursts. In these environments the dredge up of underlying heavy material into the accreted

layer affects the energetics and nucleosynthesis of the explosion.

8. Conclusions

We have presented self-consistent, numerical simulations of convective overshoot in a

2D model of the solar equatorial plane. This model employs a realistic thermal reference

state and evolves in response to convection and overshoot. We find that the convective

penetration leads to a thermal profile which is mildly subadiabatic in a shallow region at

the base of the convection zone. Beneath this mildly subadiabatic region the profile drops

precipitously to the extreme subadiabaticity of the radiative interior. Overshooting plumes

cause a mild heating in the overshoot region, leading to a region which has a slightly higher

temperature and a greater subadiabaticity than the standard solar model. The region of

increased temperature is at the location and in the sense of the helioseismology-standard

solar model sound speed discrepancy and may provide an alternative explanation for that

difference.

Passive tracer particles mix only slightly below the convective-radiative interface over

the two turnover times followed. Over longer times, a fully mixed region can be expected

at least down to a radius of 0.684 R. At this radius, the Lithium burning timescale is 6.5

billion years, a factor of 4 to 5 too long. However, only two turnover times were simulated

with tracer particles and its possible that given longer integration time particles initiated in

the convection zone could plunge deeper, thus leading to a shorter burning time. In order

to study this further, longer time integrations are needed. Furthermore, studies varying the

– 16 –

number of particles to determine the minimum radius at which there cease to be particles

should be conducted.

We should bare in mind that these results are 2D and far from the turbulent nature of

the Sun. It is unclear what three dimensional effects would be. While it seems clear that

3D models have smaller filling factors, it is unclear what this affect will be and it depends

not only on the filling factor but the number of plumes accross which that filling factor is

distributed. Furthermore, it is not clear what the effect of meridional circulation or the

full Coriolis force would be on the penetration depth and mixing. Studies on the effect

of Rayleigh number (turbulence level) on the penetration depth indicate that increasing

the Rayleigh number decreases the penetration depth (Brummell et al. 2002; Rogers &

Glatzmaier 2005a). However, it is unclear how these parametrized results come into play

in more realistic models. The jury is still out and awaits more sophisticated, 3D numerical

simulations.

We would like to thank P.Garaud, D.O. Gough, J.C. Dalsgaard, Keith MacGregor,

N. Brummell and R. Rosner for their guidance and thoughtful insight. T.R. would like

to thank the NPSC for a graduate student fellowship. Support has also been provided by

NASA SHP04-0022-00123, NASA NAG5-11220 and DOE DE-FC02-01ER41176. Computing

resources were provided by NAS at NASA Ames and by an NSF MRI grant AST-0079757.

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This preprint was prepared with the AAS LATEX macros v5.0.

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Fig. 1.— Snapshot of the temperature perturbation, representing the full computational

domain. Dark red/white represent cold/hot perturbations with respect to teh background

temperature. The convection region is dominated by descending plumes which overshoot

into the radiative region, finding themselves hotter than their surroundings (white spots at

base of convection zone). Gravity waves are generated by these overshooting plumes.

– 19 –

Fig. 2.— Time and longitudinal average of the kinetic energy density (a), vertical kinetic

energy flux divided by the reference state diffusive heat flux (b) and convective heat flux

divided by the reference state diffusive heat flux(c) before the reference state is updated.

The radius 0.718 R is the interface between the sub- and superadiabatic regions.

– 20 –

Fig. 3.— Reference state thermodynamic model. The solid line represents the initial profile;

dotted line represents evolved (in response to convection and overshoot) model. (a) supera-

diabaticity (positive ∆∇T , as defined in (5)) and subadiabaticity (negative ∆∇T ) in K/cm,

(b) represents the temperature in K. The radius 0.718 marks the transition from convective

to radiative zones.

– 21 –

Fig. 4.— Time and longitudinal average of the kinetic energy density (a), vertical kinetic

energy flux (b) and convective heat flux (c) after the reference state is updated. Again, the

kinetic energy flux and convective heat flux are divided by the original reference state dif-

fusive heat flux. Clearly convective motions penetrate further into the radiative region than

in figure 2. The radius 0.718 was the original interface between the sub- and superadiabatic

regions.

– 22 –

Fig. 5.— Distribution of tracer particles initially (a) and after two convective turnover times

(b) and a zoomed in region after the tracer particles have evolved (c). Red and blue particles

are initiated within the convection zone, while purple, cyan and black are initiated within

the radiative region. The yellow line represents the convective-radiative interface.

– 23 –

Fig. 6.— Number of particles as a function of time in the radiative region that were initiated

within the convection zone. Solid line represents the particles initiated at 0.833 R, while the

dotted line represents those particles initiated at 0.790 R. After two convective turnover

times the two sets of particles converge to having nearly 90 particles (18% of those initiated)

within the radiation zone in steady state.

– 24 –

Fig. 7.— (a) Number of particles as a function of time, initiated in the convection zone, that

make it to the specified radius within the radiation zone. (b) Number of particles starting

in the convection zone that make it into the radiative region (in steady state) as a function

of radius below the convection zone. The solid line in (b) represents a linear decline from

0.715R down to 0.70R and an exponential decline beneath 0.70R.

– 25 –

Fig. 8.— Number of particles as a function of time, initiated within the radiation zone (at

the specified radii, solid line initiated at 0.705 R), dotted line initiated at 0.690 R) which

make it into the convection zone. Note, none of the particles initiated at 0.675 R make it

into the convection zone during these two convective turnover times.